Geometry Is Beautiful

Posted on by
1

Header of Dimensions

Find nine 14-minute videos on geometry here. They’re breathtakingly beautiful!.

  • 1 Mapmaking: Stereographic projection of Earth
  • 9 Proving: the essence of mathematics
  • 2 M.C. Escher: Stereographic projection of the Platonic solids
  • 3 Four-dimensional polyhedra: Simplex, hypercube, 24-cell, 120-cell, 600-cell
  • 4 The three-sphere and 4D polyhedra viewed stereographically
  • 5 Complex numbers: Now a sphere is a “complex projective line”
  • 6 Transformations with complex numbers: Some Möbius (linear fractional) transformations, some fractals (quadratic)]
  • 7-8 Hopf fibration of three-sphere

Each video starts very simply and gets harder, but the computer graphics are so incredibly stunning that you will stay for the pictures even after you lose the narration. They are somewhat cumulative. Don’t jump into the middle. Number 9 can be watched at any time. The authors recommend last or second.

[Posted & edited by Dr. Gene Chase. Reviewed in American Mathematical Monthly, vol. 120, no. 3, March 2013, pp. 288-290.]

Women & Math

Posted on by
2

Here’s a thoughtful TED talk from Laura Overdeck of bedtime math. I’ve highlighted this website before and I think it’s such a brilliant idea for kids! Laura was the lone girl in her astrophysics undergraduate studies, so she has a great vantage point from which to give this stellar talk (pun intended! :-) ).

She has lots of great points. In particular, I liked these three simple recommendations for women–and for everyone:

  1. Don’t play the lottery.
  2. Don’t say you hate math.
  3. Don’t ask others to calculate the tip.

 

Improper integrals debate

Posted on by
1

Here’s a simple Calc 1 problem:

Evaluate  \int_{-1}^1 \frac{1}{x}dx

Before you read any of my own commentary, what do you think? Does this integral converge or diverge?

image from illuminations.nctm.org

Many textbooks would say that it diverges, and I claim this is true as well. But where’s the error in this work?

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_a^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(a)\right]=\boxed{0}

Did you catch any shady math? Here’s another equally wrong way of doing it:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_{2a}^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(2a)\right]=\boxed{\ln{\frac{1}{2}}}

This isn’t any more shady than the last example. The change in the bottom limit of integration in the second piece of the integral from a to 2a is not a problem, since 2a approaches zero if a does. So why do we get two values that disagree? (In fact, we could concoct an example that evaluates to ANY number you like.)

Okay, finally, here’s the “correct” work:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^-}\left[\int_{-1}^{a}\frac{1}{x}dx\right]+\lim_{b\to 0^+}\left[\int_b^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^-}\left[\ln|a|\right]+\lim_{b\to 0^+}\left[-\ln|b|\right]

But notice that we can’t actually resolve this last expression, since the first limit is \infty and the second is -\infty and the overall expression has the indeterminate form \infty - \infty. In our very first approach, we assumed the limit variables a and b were the same. In the second approach, we let b=2a. But one assumption isn’t necessarily better than another. So we claim the integral diverges.

All that being said, we still intuitively feel like this integral should have the value 0 rather than something else like \ln\frac{1}{2}. For goodness sake, it’s symmetric about the origin!

In fact, that intuition is formalized by Cauchy in what is called the “Cauchy Principal Value,” which for this integral, is 0. [my above example is stolen from this wikipedia article as well]

I’ve been debating about this with my math teacher colleague, Matt Davis, and I’m not sure we’ve come to a satisfying conclusion. Here’s an example we were considering:

If you were to color in under the infinite graph of y=\frac{1}{x} between -1 and 1, and then throw darts at  the graph uniformly, wouldn’t you bet on there being an equal number of darts to the left and right of the y-axis?

Don’t you feel that way too?

(Now there might be another post entirely about measure-theoretic probability!)

What do you think? Anyone want to weigh in? And what should we tell high school students?

.

**For a more in depth treatment of the problem, including a discussion of the construction of Reimann sums, visit this nice thread on physicsforums.com.

Lego Price Statistics

Posted on by
1

Do you ever get the feeling that Lego Bricks are becoming more expensive? When we were kids, boy, it felt like they were cheaper, right? I mean, the biggest sets were $150 at most. I have a HUGE Lego collection, and it definitely seems like Legos back in my day were more affordable.

Trouble is, that’s not really true. It turns out that Lego bricks have actually gotten cheaper, by almost every measure you can think of (weight/number of pieces/licensed sets). Check out this incredibly thorough post on Lego Price statistics over time. The article is entitled, “What Happened with LEGO” by Andrew Sielen. It’s very thoughtfully done.

[ht: Gene Chase]

Half-your-age-plus-seven rule

Posted on by
Reply

Looking for a great application of systems of linear inequalities for your Algebra 1 or 2 class? Look no further than today’s GraphJam contribution:

You might just give this picture to students and ask THEM to come up with the equations of the three lines.

There’s also a nice discussion to be had here about inverse functions, or about intersecting lines. And there might also be a good discussion about the domain of reasonableness.

Here are the three functions:

f_{\text{blue}}(x)=x

f_{\text{red}}(x)=\frac{1}{2}x+7

f_{\text{black}}(x)=2x-14

This is especially interesting because I never think of the rule as putting boundaries on a person’s dating age range. Usually people talk about it in the context of “how old of a person can I date?” not “how young of a person can I date?” Or rather, if you’re asking the second question, it’s usually phrased “how young of a person can date me?” (All of these questions relate to functions and their inverses!) But in fact, the half-your-age-plus-seven rule puts a lower and and upper bound on the ages of those you can date.

As far as reasonableness, is it fair to say that my daughter who is 1 can date someone who is between the age of -12 and 8.5? I don’t think so! I’m definitely going to be chasing off those -12 year-olds, I can already tell :-) .

For my daughter, the domain of reasonableness might be x\geq 18!